SWIFT J164449.3+573451: A PLUNGING EVENT WITH A POYNTING-FLUX-DOMINATED OUTFLOW

Swift J164449.3+573451 (Sw J1644+57) triggered the Swift Burst Alert Telescope on 2011 March 28 (Cummings et al. 2011; Burrows et al. 2011). As revealed by late optical observations, this transient lay at the center of a galaxy at a redshift z = 0.3534 (Levan et al. 2011a, 2011b). In the first few weeks, the average isotropic luminosity in the 0.3–10 keV band was about 10⁴⁷–10⁴⁸ erg s⁻¹. Several months later, it was still about a few × 10⁴⁵ ergs⁻¹, well above the Eddington limit. The X-ray emission declined as L_X∝t^−5/3 during the time interval from 10⁵ s to 10⁶ s after the trigger, while the subsequent decline can be approximated as L_X∝t^−4/3 (Levan et al. 2011a, 2011b; Bloom et al. 2011; Cannizzo et al. 2011).

The super-long duration (>8 months) of the X-ray activities essentially rules out gamma-ray burst (GRB) models (Shao et al. 2011). Instead it strongly favors the model of tidal disruption of a (giant) star by a massive black hole (BH; Bloom et al. 2011; Burrows et al. 2011; Levan et al. 2011b; Cannizzo et al. 2011; Shao et al. 2011; Wang & Cheng 2012). As for the central BH, it is impossible to measure the mass directly. Indirect constraints indicate that the mass should be in the range 10⁶ M_☉–10⁷ M_☉ (detailed analysis can be seen in Cannizzo et al. 2011). In the tidal disruption scenario, if a star passes within the disruption radius R_T ≈ R_*(M_BH/M_*)^1/3, the BH's tidal gravity exceeds the star's self-gravity and consequently the star is disrupted, where R_* (M_*) is the radius (mass) of the disrupted star and M_BH is the mass of the BH. After about hours to weeks, part of the remnants remain on bound and will return to the pericenter of the orbit where the material starts to be accreted inward, releasing a flare of energy (Rees 1988, 1990; Phinney 1989; Strubbe & Quataert 2009).

Theoretical calculations of the tidal disruption events suggest that the immediately accreted unbinding gas falls back to the pericenter at a rate of $\dot{M}_{\rm fb}\propto t^{-5/3}$ (Rees 1988; Phinney 1989), and the subsequent disk accretion follows a rate of $\dot{M}_{\rm fb}\tilde{\propto } t^{-4/3}$ if the disk is thick (Cannizzo & Gehrels 2009). So for t > 10⁵ s, the X-ray emission luminosity of Sw J1644+57 is proportional to the accretion rate, i.e., $L_{X}\propto \dot{M}_{\rm fb}$. In this work, we pay special attention on the physical implications of such a relation. The physical parameters of the disrupted star are also investigated.

There is an express limit derived in the Newtonian for the timescale of the return of the most bound stellar material to the pericenter (Rees 1988; Phinney 1989):

$$\begin{eqnarray} t_{\rm fb} &=& 2\pi GM_{{\rm BH}}(2\Delta E)^{-3/2}\nonumber\\ &=&0.048\, {\rm yr}\left(\frac{M_{{\rm BH}}}{5\times 10^6\,M_{\odot }}\right)^{1/2}\left(\frac{M_{\ast }}{M_{\odot }}\right)^{-1}\left(\frac{R_{\ast }}{R_{\odot }}\right)^{3/2}\mu ^3,\nonumber\\ \end{eqnarray} \tag{ 1 }$$

where ΔE = k(GM_*/R_*)(M_BH/M_*)^1/3, the dimensionless coefficient k depends on the spin-up state of the star. If the star is spun up to the break-up spin angular velocity, we have k ≈ 3. If the spin-up effect is negligible then we have k ≈ 1 (Rees 1988; Ayal et al. 2000). The dimensionless coefficient μ ≡ R_p/R_T is taken to be a free parameter in the following discussion, where R_p is the periastron distance of the star.

The star could be spun up via tidal interaction. In linear perturbation theory, the spin-up angular velocity is given by (Press & Teukolsky 1977; Alexander & Kumar 2001; Alexander & Livio 2001; Li et al. 2002)

$$\begin{equation} \frac{\omega _{s}}{\omega _{p}}\approx \frac{T_{2}(\mu ^{3/2})}{2I\mu ^{3}}, \end{equation} \tag{ 2 }$$

where ω_p ≡ v_p/r_p is the orbit angular velocity of the star at the pericenter, I is the stellar momentum of inertia in units of M_*R²_*, and T₂ is the second tidal coupling coefficient and depends on the structure of the star and the eccentricity of the orbit. For an n = 1.5 polytrope star of mass 0.76 M_☉ and radius 0.75 R_☉, Alexander & Kumar (2001) found I ≈ 0.21 and T₂(1) ≈ 0.36, corresponding to ω_s/ω_p ≈ 0.86 for μ = 1. Furthermore, they showed that the numerical simulations including nonlinear effects led to a larger energy transfer from the orbit to the star and a larger spin-up than that predicted by linear theory. Therefore, we take k = 3 as the fiducial value in our analysis. For completeness we also present the results in the non-spinning case (i.e., k = 1).

The bound material returns to pericenter at the rate (Phinney 1989)

$$\begin{eqnarray} \dot{M}_{\rm fb} &=& \frac{2\Delta M}{3t_{\rm fb}}\left(\frac{t-t_{s}}{t_{\rm fb}}\right)^{-5/3}\nonumber\\ &=& 5.6\times 10^{23}\ {\rm g\ s^{-1}}\left(\frac{f}{0.1}\right)\left(\frac{M_{{\rm BH}}}{5\times 10^6\,M_{\odot }}\right)^{1/3}\nonumber\\ &&\times \left(\frac{M_{\ast }}{M_{\odot }}\right)^{1/3}\left(\frac{R_{\ast }}{R_{\odot }}\right)\left(\frac{t-t_{s}}{1\,{\rm yr}}\right)^{-5/3}\mu ^2, \end{eqnarray} \tag{ 3 }$$

where t_s is the time of initial tidal disruption, ΔM is the mass that falls back to pericenter, and the dimensionless factor f is defined as f ≡ ΔM/M_*.

When the accretion rate of the fall-back material is highly super-Eddington, only a fraction (1 − f_out) of such material forms a disk and can be accreted all the way down to the central BH, i.e., $\dot{M}=(1-f_{{\rm out}})\dot{M}_{{\rm fb}}$. The remaining part will instead leave the system, undergoing a strong radiation pressure. Strubbe & Quataert (2009) took a constant f_out = 0.1. However, numerical simulation indicates that the parameter f_out is a growing function of $\dot{M}_{{\rm fb}}/\dot{M}_{{\rm Edd}}$, reaching f_out ≈ 0.7 for $\dot{M}_{{\rm fb}}/\dot{M}_{{\rm Edd}}=20$ (Dotan & Shaviv 2010). Considering the observed X-ray luminosity L_X ∼ 10⁴⁷–4 × 10⁴⁸ erg s⁻¹ during the first 10⁶ s after the trigger (Burrows et al. 2011), even for a radiation efficiency as high as 0.1, we need an accretion rate $\dot{M}=10L_{X}/c^2\sim 5\times 10^{-7}\hbox{--}2\times 10^{-5}\,\dot{M}_{\odot }$. The Eddington luminosity can be scaled as L_Edd ≃ 6.25 × 10⁴⁴M_6.7 erg s⁻¹, so the Eddington rate $\dot{M}_{{\rm Edd}}\equiv 10 L_{{\rm Edd}}/c^2=3.5\times 10^{-7}\,M_{\odot }\, {\rm s^{-1}}$. We then have $\dot{M}/\dot{M}_{{\rm Edd}}\sim 1\hbox{--}60$. We define a free dimensionless parameter ξ = f(1 − f_out), the fraction of the material that is actually accreted onto the central BH.

Assuming that the jet radiation efficiency is during the stellar debris fall-back accretion, the intrinsic jet luminosity then can be given by

$$\begin{eqnarray} L_{\rm j}&=&\epsilon \dot{M}c^2\approx 2.5\times 10^{42}\ {\rm erg \ s^{-1}} \left(\frac{\epsilon }{0.01}\right)\left(\frac{\xi }{0.05}\right)\nonumber\\ &&\times \left(\frac{M_{{\rm BH}}}{5\times 10^6\,M_{\odot }}\right)^{1/3}\! \left(\frac{M_{\ast }}{M_{\odot }}\right)^{1/3}\! \left(\frac{R_{\ast }}{R_{\odot }}\right)\!\left(\frac{t-t_{s}}{1\,{\rm yr}}\right)^{-5/3}\mu ^2,\nonumber\\ \end{eqnarray} \tag{ 4 }$$

where the efficiency is normalized to ∼ 0.01 (in Section 3.3, we will show that the jet should be launched mainly via magnetic activities (e.g., the Blandford–Znajek (BZ) effect) and the efficiency is about 0.01). As shown below, the conclusion drawn in this section is independent of the value of .

When most of bound debris falls back to the pericenter, the jet luminosity peaks at t − t_s = t_fb and can be estimated as

$$\begin{eqnarray} L_{\rm j,peak} &\approx& 3.9\times 10^{44}\ {\rm erg\ s^{-1}}\left(\frac{\epsilon }{0.01}\right)\left(\frac{\xi }{0.05}\right)\nonumber\\ &&\times \left(\frac{M_{{\rm BH}}}{5\times 10^6\,M_{\odot }}\right)^{-1/2} \left(\frac{M_{\ast }}{M_{\odot }}\right)^{2}\left(\frac{R_{\ast }}{R_{\odot }}\right)^{-3/2}\mu ^{-3}.\nonumber\\ \end{eqnarray} \tag{ 5 }$$

The observed maximal X-ray luminosity is ∼4 × 10⁴⁸ erg s⁻¹. For a collimated emitting region with a half-opening angle θ_j, we have the constraint L_{X, peak} ⩾ 2 × 10⁴⁶(θ_j/0.1)² erg s⁻¹. Assuming that most of the radiated energy is in the X-ray band during the time interval, we have

$$\begin{eqnarray} &&\left(\frac{\epsilon }{0.01}\right)\left(\frac{\theta _{j}}{0.1}\right)^{-2}\left(\frac{\xi }{0.05}\right)\left(\frac{M_{{\rm BH}}}{5\times 10^6\,M_{\odot }}\right)^{-1/2}\nonumber\\ &&\quad \times \left(\frac{M_{\ast }}{M_{\odot }}\right)^{2}\left(\frac{R_{\ast }}{R_{\odot }}\right)^{-3/2}\mu ^{-3}\ge 51. \end{eqnarray} \tag{ 6 }$$

The observed X-ray fluence S_X (0 s < t < 10⁷ s) suggests a total energy ΔE_X = θ²_j∫^t₀L_X(t)dt/2 ∼ 1 × 10⁵¹(θ_j/0.1)² erg. The total mass of the accreted material is thus

$$\begin{equation} M_{\ast }=\frac{\Delta E}{\xi \epsilon c^2}\approx \frac{\Delta E_{X}}{\xi \epsilon c^2 }\approx 1.1\,M_{\odot }\left(\frac{\epsilon }{0.01}\right)^{-1}\left(\frac{\xi }{0.05}\right)^{-1}\left(\frac{\theta _{j}}{0.1}\right)^2. \end{equation} \tag{ 7 }$$

The approximated mass–radius relationship can be scaled as (R_*/R_☉) = (M_*/M_☉)^η. For the main-sequence stars, we have η ≈ 0.8 for 0.1 M_☉ < M_* < 1 M_☉ and η ≈ 0.6 for 1 M_☉ < M_* < 10 M_☉ (Kippenhahn & Weigert 1994). With Equations (6) and (7), we obtain

$$\begin{equation} \left(\frac{M_{\ast }}{M_{\odot }}\right)^{1-3\eta /2}\mu ^{-3}\left(\frac{M_{{\rm BH}}}{5\times 10^6\,M_{\odot }}\right)^{-1/2}\ge 46.4, \end{equation} \tag{ 8 }$$

which then yields

$$\begin{equation} \mu \le 0.36\left(\frac{M_{{\rm BH}}}{5\times 10^6\,M_{\odot }}\right)^{-1/6}\left(\frac{M_{\ast }}{M_{\odot }}\right)^{1/3-\eta /2}. \end{equation} \tag{ 9 }$$

Interestingly, the parameter μ is independent of , ξ, and θ_j. Its dependence on both the stellar mass and the BH mass is also very weak. The above analysis is under the condition that the star is spun-up to the break-up spin angular velocity, i.e., k = 3. For the non-spinning case k = 1, we can obtain a more stringent result μ ⩽ 0.16((M_BH)/(5 × 10⁶ M_☉))^−1/6(M_*/M_☉)^{1/3 − η/2}. Therefore, we conclude that the periastron distance is likely well within the tidal disruption radius (i.e., it is a plunging event), in agreement with Cannizzo et al. (2011).

Based on the observation data, the peak accretion rate can be derived with Equations (5) and (7),

$$\begin{eqnarray} \dot{M}_{\rm peak}&=&\frac{ L_{\rm j, peak}}{\epsilon c^2}=\frac{\xi M_{\ast }L_{\rm X,peak}}{\Delta E_{X}}\nonumber\\ &=& 4.4\times 10^{-5}\,M_{\odot } s^{-1}\left(\frac{\xi }{0.05}\right)\left(\frac{M_{\ast }}{1.1\,M_{\odot }}\right). \end{eqnarray} \tag{ 10 }$$

The accretion rate $\dot{M}$ can be scaled as $\dot{M}=\dot{M}_{\rm peak}((t- t_{s})/{1yr})^{-5/3}$ during the fall-back accretion process.

In this plunging event, the disrupted star's orbit is likely to misalign with the equatorial plane of the spinning central BH. A tilted accretion disk should be formed and the jet aligned with the disk normal vector is expected to precess (Stone & Loeb 2012; Lei et al. 2012). Saxton et al. (2012) have analyzed the X-ray timing and spectral variability of Sw J1644+57 and found periodic modulation, possibly due to jet precession.

After the time t ∼ 10⁵ s, the observed X-ray luminosity followed the fall-back accretion rate (Levan et al. 2011a, 2011b; Bloom et al. 2011), i.e., $L\propto \dot{M}$. Such a relationship has shed some light on the underlying physics.

3.1. Thermal X-Ray Radiation from the Disk?

While the fall-back accretion rate is super-Eddington, the stellar material returning to pericenter is so dense that it cannot radiate and cool. The gas is most likely to form an advective-dominated accretion flow accompanied by powerful outflow, which dominates the emission. Most of the radiation will be emitted from the outflow's photosphere. When the photosphere lies inside the outflow, the photosphere's radius and temperature can be written as (Strubbe & Quataert 2009; Rossi & Begelman 2009)

$$\begin{equation} R_{{\rm ph}}\sim 4f_{{\rm out}}f_{v}^{-1}\left(\frac{\dot{M}_{{\rm fb}}}{\dot{M}_{{\rm Edd}}}\right)R_{p,3R_{s}}^{1/2}R_{s} \end{equation} \tag{ 11 }$$

and

$$\begin{equation} T_{{\rm ph}}\sim 1\times 10^{5}\ {\rm K}\, f_{{\rm out}}^{-1/3}f_{v}^{1/3}\left(\frac{\dot{M}_{{{\rm fb}}}}{\dot{M}_{{{\rm Edd}}}}\right)^{-5/12} M_{6.7}^{-1/4}R_{p,3R_{s}}^{-7/24}, \end{equation} \tag{ 12 }$$

where R_s ≡ (2GM_BH/c²) is the Schwarzschild radius and f_v is the ratio of the terminal velocity of gas with the escape velocity at a radius of ∼2R_p. However, the photons escape from the photosphere are mainly in the UV optical band and have a blackbody spectrum. The UV optical emission luminosity can be scaled as $\nu L_{\nu }\sim 4\pi R_{{\rm ph}}^2\nu B_{\nu }(T_{{\rm ph}})\propto R_{{\rm ph}}^2 T_{{\rm ph}} \propto \dot{M}_{{{\rm fb}}}^{19/12}$.

These optical photons could be Compton-scattered by the relativistic electrons in the outflow. The energy of the photons getting scattered is hν_IC ≈ D²γ²hν, where D = 1/[Γ(1 − βcos θ)] is the Doppler factor, Γ is the Lorentz factor of the outflow, θ is the angle between the outflow axis and the observer's line of sight, and γ is the Lorentz factor of the relativistic electrons. The energy of inverse-Compton-scattered photons can peak in the X-ray band if the parameter D ∼ 1 and γ ∼ 10. However, even in this case, the X-ray luminosity does not satisfy the relation $L_{x}\propto \dot{M}_{{\rm fb}}$, which is inconsistent with the observational data.

3.2. Neutrino-annihilation-launched Outflow?

When the mass accretion rate is high enough, the accretion proceeds via neutrino cooling and neutrinos can carry away a significant amount of energy from the inner regions of the disk. This mechanism is used to explain the launch of at least some GRB outflows (Popham et al. 1999; Fan & Wei 2011; Fan et al. 2012). The luminosity may be well approximated by a simple formula (Zalamea & Beloborodov 2011):

$$\begin{eqnarray} L_{\nu \bar{\nu }} & \approx & 1.1\times 10^{52}\,\chi _{\rm ms}^{-4.8}\, \left(\frac{M_{{\rm BH}}}{3M_\odot }\right)^{-3/2}\\ \ms && \times \left\lbrace \begin{array}{@{}ll@{}}0 & \dot{M}<\dot{M}_{{\rm ign}} \\ \dot{m}^{9/4} \;\; & \dot{M}_{{\rm ign}}<\dot{M}<\dot{M}_{{\rm trap}} \\ \dot{m}_{\rm trap}^{9/4}\;\; & \dot{M}>\dot{M}_{{\rm trap}} \\ \end{array} \right\rbrace \,{\rm erg\ s}^{-1}, \end{eqnarray} \tag{ 13 }$$

where $\dot{m}=\dot{M}/M_{\odot }$ s⁻¹, χ_ms = R_ms(a)/R_s, R_ms is the radius of the marginally stable orbit, $\dot{M}_{{\rm ign}}$ is the mass accretion rate for igniting neutrino emission, and $\dot{M}_{{\rm trap}}$ is the mass accretion rate when the emitted neutrino becomes trapped in the disk and advected into the BH. The characteristic accretion rates $\dot{M}_{{\rm ign}}$ and $\dot{M}_{{\rm trap}}$ depend on the viscosity parameter α and the mass of the central BH. Based on the work of Beloborodov (2003), for ν-transparent, the accretion rate should be as large as $\dot{M}_{{{\rm ign}}}>7.6\times 10^{30}({r}/{3r_{s}})^{1/2}({\alpha }/{0.1}) ({M_{{\rm BH}}}/{M_{\odot }})^2\ {\rm g\ s^{-1}}$. For the event Sw J1644+57, which is a plunging one, the peak accretion rate is $\dot{M}_{\rm peak}\approx 4.4\times 10^{-5}\,M_{\odot }\, {\rm s^{-1}}$, which is far less than $\dot{M}_{{\rm ign}}$. Hence we conclude that the observed X-ray emission could not be produced by neutrino annihilation (see Shao et al. 2011 for an alternative argument disfavoring a neutrino mechanism).

3.3. Poynting-flux-dominated Outflow?

Extracting energy from the rotating BH may be possible through the BZ mechanism (Blandford & Znajek 1977). Such a process is based on the expectation that the differential rotation of the disk will amplify pre-existing magnetic fields until they approach equipartition with the gas kinetic energy. For a BH of mass M_BH and angular momentum J, with a magnetic field B_⊥ normal to the horizon at R_h, the power arising from the BZ mechanism is given by (e.g., Thorne et al. 1986)

$$\begin{equation} L_{{\rm BZ}}=\frac{\pi }{8}\omega _{F}^2\left(\frac{B_{\bot }^{2}}{4\pi }\right)R_{h}^2c\left(\frac{J}{J_{{\rm max}}}\right)^2, \end{equation} \tag{ 15 }$$

where J_max = GM²/c is the maximal angular momentum of the BH. The factor ω²_F = Ω_F(Ω_h − Ω_F)/Ω²_h depends on the angular velocity of field lines Ω_F relative to that of the BH, Ω_h. Usually we adopt ω_F = 1/2, which maximizes the power output (Macdonald & Thorne 1982; Thorne et al. 1986). We follow the common assumption that the magnetic field in the disk will rise to some fraction of its equipartition value P_mag = (B²/8π) ∼ αP in the inner disk. The pressure P = ρc²_s is given by (Armitage & Natarajan 1999)

$$\begin{equation} P=\frac{\sqrt{2}\dot{M}}{12\pi \alpha }(5+2\varepsilon)^{1/2}(GM)^{1/2}R^{-5/2}, \end{equation} \tag{ 16 }$$

where ε is the parameter governing the property of the disk. For a thick disk we have ε < 1 otherwise ε > 1. In the inner region of the disk, we assume B_⊥ ≈ B, R ≈ R_h = GM/c². The BZ power for the case of a maximally rotating BH (i.e., J = J_max) can be estimated as

$$\begin{equation} L_{{\rm BZ}}\approx 7\times 10^{-3}(5+2\varepsilon)^{1/2}\dot{M}c^2, \end{equation} \tag{ 17 }$$

corresponding to an efficiency $\epsilon _{{\rm BZ}}=L_{{\rm BZ}}/\dot{M}c^2\sim 10^{-2}$ for the thick-disk model. In the thin-disk scenario the radiative efficiency can be as high as ∼0.1. For Sw J1644+57, at the time t − t_s = t_fb + 10⁶ s, the mass accretion rate is about 3.3 × 10⁻⁸ M_☉ s⁻¹, and the observed luminosity is 2L_BZθ⁻²_j ∼ 5 × 10⁴⁷ erg s⁻¹(θ_j/0.1)⁻², consistent with the observation.

Our conclusion that the outflow powering the super-long X-ray emission should be launched via magnetic activities (e.g., BZ mechanism) is consistent with that of Shao et al. (2011). Lei & Zhang (2011) have also analyzed the jet launched by the BZ mechanism and then constrained the physical parameter of the central BH. One interesting finding is that the central BH should have a moderate to high spin.

In the Poynting-flux-dominated outflow, the X-ray emission could be due to the dissipation of the magnetic field (Usov 1994; Thompson 1994). There are several magnetic field dissipation models that could produce the observed emission, such as the global MHD condition breakdown model (Usov 1994), the gradual magnetic reconnection model (Drenkhahn & Spruit 2002), the magnetized internal shock model (Fan et al. 2004), and the collision-induced reconnection model (Zhang & Yan 2011). For illustration, here we take the global MHD condition breakdown model to calculate the emission. By comparing with the pair density (∝r⁻², r is the radial distance from the central source) and the density required for co-rotation (∝r⁻¹ beyond the light cylinder of the compact object), one can estimate the radius at which the MHD condition breaks down, which reads (Usov 1994; Zhang & Mészáros 2002)

$$\begin{equation} r_{{\rm MHD}}\sim 5\times 10^{20}\ {\rm cm}\left(\frac{L}{10^{47}}\right)^{1/2} \left(\frac{\sigma }{10}\right)^{-1}\left({\frac{t_{v,m}}{10^2}}\right) \left(\frac{\Gamma }{10}\right)^{-1}, \end{equation} \tag{ 18 }$$

where σ is the ratio of the magnetic energy flux to the particle energy flux, Γ is the bulk Lorentz factor of the outflow, and t_{v, m} is the minimum variability timescale of the central engine. Beyond this radius, intense electromagnetic waves are generated and outflowing particles are accelerated (e.g., Usov 1994). Such a significant magnetic dissipation process converts the electromagnetic energy into radiation.

At r_MHD, the corresponding synchrotron radiation frequency can be estimated as (Fan et al. 2005; Gao & Fan 2006)

$$\begin{eqnarray} &&\nu _{m,{\rm MHD}}\sim 1.5\times 10^{18}\ {\rm Hz}\left(\frac{1+z}{1.35}\right)^{-1}\left(\frac{\zeta }{0.1}\right)\nonumber\\ &&\quad \times C_{p}^{2}\left(\frac{\sigma }{10}\right)^3\left(\frac{\Gamma }{10}\right)\left(\frac{t_{v,m}}{10^2}\right), \end{eqnarray} \tag{ 19 }$$

where C_p ≡ (_e/0.1)[13(p − 2)/3(p − 1)], _e is the fraction of the dissipated comoving magnetic field energy converted to the comoving kinetic energy of the electrons, the accelerated electrons distribute as a single power-law dn/dγ_e∝γ^−p_e, and ζ < 1 reflects the efficiency of magnetic energy dissipation. Most of the energy is radiated in the X-ray band.

The observations of Sw 1644+57 suggest that the long-lasting fall-back accretion onto a BH can produce energetic X-ray emission and the radiation luminosity traces the accretion rate (i.e., $L_{X}\propto \dot{M}$). One interesting question is whether or not a similar process takes place in GRBs. The answer may be yes. Here, we only discuss the X-ray steep decline (quicker than t⁻³; see Figure 1 of Zhang et al. 2006 for illustration) following the prompt emission in GRBs. In the collapsar model, a fraction of the gas in the core of the collapsing star does not have sufficient centrifugal support and directly forms a central BH. The rest of the material will have sufficient angular momentum to go into orbit around the BH. The fall-back accretion rate is tightly related to the pre-collapse stellar density profile, which is of the form ρ∝r^−τ. A numerical simulation has revealed that when the outermost 0.5 M_☉ layer of the star (where τ > 5) is accreted, the fall-back accretion rate can reach a value of $\dot{M}_{{{\rm fb}}}\propto t^{-3}$ or steeper (MacFadyen et al. 2001; Kumar et al. 2008). If the relation $L_{X}\propto \dot{M}$ still holds, one has an X-ray emission decline steeper than t⁻³, in agreement with observational data.¹

Swift J164449+573451 is a peculiar outburst which is most-likely powered by the tidal disruption of a star by a massive BH. In this work, we find that the ratio of the periastron distance to the disruption radius μ < 1, implying that SW J1644+77 is a plunging event (see Section 2). The mass of the plunging star, however, cannot be tightly constrained due to its strong dependence on the poorly understood parameters (the radiation efficiency), ξ (the fraction mass of the star that is eventually accreted into the central), and θ_j (the half opening angle of the collimated outflow).

As a tidal disruption event, the accretion rate $\dot{M}$ at late times (say, t > 10 day) is relatively well understood and is widely believed to be ∝t^−4/3. The detected X-ray emission L_X shows a rather similar decline behavior. Since the forward shock origin of the long lasting and highly variable X-ray emission has already been convincingly ruled out (Shao et al. 2011), the X-ray emission has to be from an outflow launched by the accreting BH. These two facts strongly suggest that $L_{X}\propto \dot{M}$, which can shed valuable light on the underlying physics, in particular, the energy extraction process. Three kinds of possible mechanisms have been examined and only the Poynting-flux-dominated outflow model is found to be able to account for the data (see Section 3 for details). Therefore, the magnetic activity at the central engine (e.g., Blandford–Znajek process or Blandford–Payne process) plays the main role in extracting the rotation energy of the BH and then launching the outflow. We suggest that $L_{X}\propto \dot{M}$ may also hold in the quick decline phase of GRBs.

We thank the anonymous referee for constructive comments and Dr. Yizhong Fan for his kind help in improving the presentation. This work was supported in part by the National Natural Science Foundation of China under the grant 11073057 and by the Key Laboratory of Dark Matter and Space Astronomy of Chinese Academy of Sciences.